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Finite structures, their theory, their construction, and applications. (Endliche Strukturen, ihre Theorie, ihre Konstruktion and Anwendungen.) (German) Zbl 0884.05025
The author gives several examples how the theory of finite structures can constructively be applied to combinatorial problems. First he roughly sketches a proof of Cayley’s result that the number of labelled trees with \(n\) points is \(n^{n-2}\), based on a proof of A. Joyal. In the second part the author deals with unlabelled structures. He states the so-called fundamental lemma relating sets of group orbits to certain double cosets and transversals. This is applied to examples: (1) Chemical isomeres, here \(\text{C}_{12} \text{O}_2 \text{H}_4 \text{Cl}_4\) (dioxin) is considered in some detail. (2) The number of unlabelled graphs with a given number of points and edges. (3) The number of isometric classes of \((n,k)\)-codes, thereby using the wreath product of groups. (4) Construction of the first 7-designs thereby using the Kramer/Mesner matrix. Finally a method to gain transversals of double cosets developed by B. Schmalz is sketched.
This paper is a written and extended version of a talk given by the author at the annual conference of the DMV in Jena. Somebody not really familiar with the subject would want to consider the authors book mentioned in the list of references to gain a deeper understanding.
MSC:
05B30 Other designs, configurations
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
05C90 Applications of graph theory
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